Existence of travelling-wave solutions and local well-posedness of the Fowler equation

We study the existence of travelling-waves and local well-posedness in a subspace of $C_b^1(\mathbb{R})$ for a nonlinear evolution equation recently proposed by Andrew C. Fowler to study the dynamics of dunes.


Introduction
1.1. General setting. Dunes are land formations of sand which are subject to different forms and sizes based on their interaction with the wind or water or some other mobile medium. In the case of dunes in the desert their shapes depend mainly on the amount of sand available and on the change of the direction of the wind with time (see Herrmann and Sauermann [6]). Some examples of dune patterns are longitudinal, transverse, star and Barchan dunes, however, there are more than 100 categories of dunes. Dunes also occur under rivers, for similar reasons, but their shapes are less exotic in this case, because the flow is mainly uni-directional.
An interesting topic is to try to understand if the shape of a dune is maintained when it moves. With regard to Barchan dunes, for example, Herrmann and Sauermann [6] have given some arguments against the hypothesis that Barchan dunes are solitary waves, mainly because they constantly lose sand at the two horns and tend to disappear if not supplied with new sand. Recently, Durán, Schwämmle and Herrmann [2] considered a minimal model for dunes consisting of three coupled equations of motion to study numerically the mechanisms of dune interactions for the case when a small Barchan dune collides with a bigger one; four different cases were observed, depending only on the relative sizes of the two dunes, namely, coalescence, breeding, budding, and solitary wave behavior.
In this paper, we are concerned with the following evolution equation proposed by Fowler (see [3], [4] and [5] for more details) to study nonlinear dune formation: where u = u(x, t) represents the dune amplitude, x ∈ R, and t ≥ 0. The second and fourth terms of equation (1.1) correspond to the nonlinear and nonlocal terms respectively, while the third term is the dissipative term. Let us give a brief description of the model derivation. For more details, we refer to Fowler [3,4,5], which we follow closely. The model stems from the Exner law, which is the conservation of mass for the sediment: where the bedload transport q = q(τ ) is assumed, in the case of dunes, to depend only on the stress τ exerted by the fluid on the erodible bed. We assume a twodimensional flow, where x is the horizontal direction and the second direction is the upwards coordinate orthogonal to x. This should account for transverse dunes, but obviously not for other types of dunes. The nonlocal term in equation (1.1) arises from a subtle modelling of the basal shear stress τ b . Roughly speaking, the turbulent bottom shear stress is given by τ b ≈ f ρv 2 , where ρ is the fluid density, f is a dimensionless friction coefficient and v is the mean fluid velocity (vertically averaged). By performing an asymptotic expansion with respect to the aspect ratio ǫ of the evolving bedform, ǫ = bed thickness fluid depth ≪ 1, and a perturbation analysis of a basic Poiseuille flow (Orr-Sommerfeld equation), Fowler [3,4,5] was able to obtain the following expression: where α is a positive constant proportional to Re 1/3 , Re being the Reynolds number. Due to the bed slope ∂u ∂x , there is an additional force generated by gravity g. Therefore, the net stress causing motion is actually τ = τ b − (ρ s − ρ)gD s ∂u ∂x , where ρ s is the sediment density and D s the mean diameter of a sediment particle. As long as u is small, the shallow water approximation applies to the velocity v and, for small Froude number, the (dimensionless) mean fluid velocity can be approximated by v ≈ 1 1−u . Thus, the mean fluid velocity and the bottom shear stress depend on the motion of the dune profile u, and therefore there is a feedback between the dune profile and the motion of the fluid. In dimensionless variables, taking all physical constants equal to 1, the resulting net stress is then given by Notice that the nonlinear nonlocal term 2u +∞ 0 ξ −1/3 ∂u ∂x (x − ξ, t)dξ has been discarded. By a Taylor expansion, up to order 2, we get q(τ ) ≈ q(1) + q ′ (1)(τ − 1) + 1 2 q ′′ (1)(τ − 1) 2 . Now, considering a moving spatial coordinate, i.e. replacing x by the new variable x − q ′ (1)t, plugging q into the Exner equation, after a suitable rescaling, we obtain the canonical equation (1.1).
Some numerical computations have been performed by Fowler [4,5] and Alibaud, Azerad and Isèbe [1]. Fowler mentions the fact that the numerical solution, computed with a pseudo-spectral method in a large domain, starting from random initial data, converges to a final state consisting of one travelling-wave. Alibaud et al., using a finite difference scheme valid for a bounded time interval, starting from a compactly supported nonnegative initial data, showed that the numerical solution of the Fowler equation (1.1) quickly evolves to a solution with a non zero negative part, showing the erosive effect of the nonlocal term. They also establish theoretically the non monotone property of (1.1), namely the violation of the maximum principle (see also Remark 3.2 below).
To the authors' knowledge, ours is the first study to report a rigorous mathematical proof of the existence of travelling-waves for dune morphodynamics. We notice that we have not found nontrivial travelling-waves of the solitary-wave type for this model (see Remark 2.5 below), however we could not exclude the possibility that they exist. What we obtain is more bore-like travelling-waves. This type of travelling dunes has not been observed yet, to the author's knowledge. This may put under question the validity of the Fowler equation to faithfully describe dune morphodynamics. The authors hope that these results could be of interest for geographers, geologists, oceanographers and others.
1.2. Organization of the paper. In Section 2 we study the existence of travellingwave solutions to equation (1.1). The main result of this section is Theorem 2.1 which implies that for each wave speed d > 0, and η in a neighborhood of zero, η ∈ R, there exists a travelling-wave solution u(x, t) = φ(x − dt) to the following version of equation (1.1) ; the idea of its proof is to use the implicit function theorem on suitable Banach spaces. Then, by a scaling argument and considering a suitable translation of the travelling-wave, we extend this result for any η ∈ R and any wave speed d ∈ R.
Section 3 is devoted to proving local well-posedness (LWP) for the integral equation associated to the initial value problem (IVP) for equation (1.1). Inspired by the regularity of the travelling-wave obtained in Section 2, we consider a suitable subspace of C 1 b (R). The analysis of the linear equation associated to equation (1.1) is addressed in Sub-section 3.1. Next, in Sub-section 3.2, the main result of this section is stated in Theorem 3.1; it gives local-in-time existence of the solution of the integral equation associated to the IVP for equation (1.1), with initial data belonging to the subspace X of C 1 b (R), where X := {f ∈ C 1 b (R); f ′ is uniformly continuous}.

1.3.
Notations. -We denote by R and C the sets of all real and complex numbers respectively. N denotes the set of all natural numbers.
-We denote by C(c 1 , c 2 , . . .) a constant which depends on the parameters c 1 , c 2 , . . . C is assumed to be a non-decreasing function of its arguments.
-The norm of a measurable function f ∈ L p (Ω), for Ω a subset of R, is written f p L p (Ω) = Ω |f | p dx for 1 ≤ p < +∞, and f L ∞ (Ω) = ess sup Ω |f |. The inner product of two functions f, g ∈ L 2 (Ω) is written as (f, g) = Ω fḡdx. We will often omit set Ω when context is clear.
-We write C ∞ (R) to denote the space of all continuous complex-valued functions defined on R which tend to zero at infinity. -We denote by C b (R) = C 0 b (R) the space of all bounded continuous real-valued functions on R with the norm · L ∞ . Moreover, for every k ∈ N, we write -If X and Y are two Banach spaces, we denote by L(X, Y ) the set of all continuous linear mappings defined on X with values in Y ; if X = Y , we denote by L(X).

Existence of Travelling-Wave Solutions of the Fowler Equation
We begin this section with some notations and preliminary results. We define where χ A is used to denote the characteristic function of the set A. We also define We note that, since ψ ∈ S ′ (R), it follows that for φ ∈ S(R), one has that ψ * φ ∈ C ∞ (R) ∩ S ′ (R) and ψ * φ = √ 2πψφ (see Rudin [8]). Then, for ϕ ∈ S(R), Next lemma gives the Fourier transform of function ψ.
Proof. We define the function ψ n (x) := χ (0,n) (x)x −1/3 , for all x ∈ R, and n ∈ N. It is not difficult to see that ψ n → ψ in S ′ (R) as n goes to infinity. Let ϕ ∈ S(R). Then it follows that |, for all n ∈ N, and x ∈ R. Therefore, the dominated convergence theorem implies that This completes the proof of the lemma.
Remark 2.1. Let s ∈ R. If u ∈ H s (R), one can define g[u] through its Fourier transform by In this section we consider the following, more general, version of equation (1.1): where η ∈ R. We will show existence of travelling-wave solutions to equation (2.5), for any η ∈ R. First, we consider the case η = 0. For any d ∈ R (see Johnson [7]), is a solution to equation (2.5) with η = 0.
It is straightforward to check that if u is a solution to the equation then u λ satisfies equation (2.5). Hence, if φ is a travelling-wave solution of equation We define, for c ∈ R, the functions We see that g[g c ] = I 1 +I 2 , where I j := ψ j * h c for j = 1, 2, with ψ 1 := ψ·χ (0,1) , and ψ 2 := ψ·χ (1,+∞) . Now, we state some immediate properties of the function g[g c ].
a.) Let p > 3. Since ψ 1 ∈ L 1 (R), and ψ 2 ∈ L p (R), it follows from the Young inequality for convolution that In fact, it follows from the dominated convergence theorem that I 1 is continuous and I 1 (x) → 0 as |x| → ∞. Moreover, Hölder's inequality and the dominated convergence theorem imply that The continuity of I 2 is shown similarly to the continuity of I 1 .
Let c ∈ R. In the sequel, we will consider the following spaces: By integration by parts one has that (2.10) Thus, a sufficient condition to guarantee that φ satisfies the last equation is We denote by τ c the function given by We now define the function G = G c , which is well defined on R × X by Remarks 2.3 and 2.4 above, as (2.13) Hence, φ satisfies equation (2.12) if and only if ϕ verifies the equation G(η, ϕ) = 0.
The following theorem implies the existence of a travelling-wave solution, u(x, t) = φ(x − ct) with c > 0 and φ ∈X, to equation (2.5) for η in a neighborhood of zero; its proof uses the implicit function theorem.
Proof. Let c > 0. The mapping G = G c is defined on the Banach space R × X taking values in the Banach space (C b (R), · L ∞ ), and satisfies G(0, 0) = 0.
We now claim that ∂ 1 G and ∂ 2 G exist as partial F-derivatives (Fréchet derivative) on R × X and that the partial F-derivative ∂ 2 G(0, 0) : In fact, let us take (η, ϕ) ∈ R × X. One can see that (2.14) where we have used inequality (2.11). Hence, ∂ 1 G, ∂ 2 G exist as partial F-derivatives on R × X.
We will now show that the partial F-derivative ∂ 2 G(0, 0) = τ c + ∂ x : X → C b (R) is bijective. We begin with the injectivity; we emphasize here that the definition of the space X ⊂ C 1 b (R) was chosen to ensure the injectivity of the mapping ∂ 2 G(0, 0).
Let f be an element of X such that τ c f + f ′ = 0. By solving the last ordinary differential equation, one gets x .
We will now show that the mapping ∂ 2 G(0, 0) is onto. Let y be an element of C b (R). By the method of variation of parameters, we obtain that the function . We will prove that g ∈ X for a suitably chosen real number λ. First, we remark that there exists a unique λ = λ y,c ∈ R such that g ′ h ′ c dx=0. In fact take where we note that It remains to show that g given by (2.15) and (2.16) belongs to C 1 b (R). It is immediate to see that g ∈ C(R), we need to show that g is bounded. We have that Then g ∈ C b (R). Moreover, since g satisfies the equation τ c g + g ′ = y, it follows that g ∈ C 1 b (R). Hence, g ∈ X. Therefore, ∂ 2 G(0, 0) is a surjective mapping. It is not difficult to see, by using inequality (2.11), that G, ∂ 1 G and ∂ 2 G are continuous on R × X. Then, the implicit function theorem implies the first part of the theorem. Furthermore, from (2.13) one can see that function G is quadratic in ϕ and linear in η, therefore it is not difficult to verify that Finally, the second part of the theorem is then a consequence of the fact that the mapping G is a C ∞ -map on R × X.
Proof. Let c > 0. By Theorem 2.1 there exists λ 0 = λ 0 (η, c) > 0 such that for every Making x → +∞, integrating by parts, and then applying Parseval's relation and Remark 2.1, we obtain These formal steps can be justified by assuming for instance that φ ∈ H 2 (R). Thus, equation (2.17) implies that if η ≤ 0, then φ = 0. We can then conclude that there are no nontrivial travelling-waves of the solitary-wave type for equation (2.5) when η ≤ 0. However, in the physical case, that is to say when η = 1 or more generally when η > 0, equation (2.17) does not preclude the possibility that they may exist.
3. Local Theory in a subspace of C 1 b (R) In Section 2, we proved the existence of a travelling-wave solution u(x, t) = φ(x − ct) to equation (2.5) for any η ∈ R, where c is an appropriate positive number and φ ∈ C 1 b (R). Motivated by this last result, we will consider in this section the local well-posedness theory for the following initial value problem (IVP) where g[u] is given by (2.2), and u 0 belongs to a suitable subspace of C 1 b (R). The Cauchy problem associated to the IVP (3.1) for initial data u 0 ∈ L 2 (R) was recently studied by Alibaud, Azerad and Isèbe [1].
For t ≥ 0, we define the operator E(t) by where φ ∈ C b (R) (see Lemma 3.11 below). Now, we define the following spaces One can see that (Y, · C b (R) ), and (X, · C 1 b (R) ) are Banach spaces and that X ֒→ Y . In Sub-section 3.2 we will show local-in-time well-posedness of the IVP (3.1), with initial data u 0 ∈ X.
The following lemma contains a calculus result.
Lemma 3.1. Let h : R → C be a function which satisfies the following conditions: Proof. Since h ∈ L 1 (R), it follows from the Riemann-Lebesgue lemma thatĥ ∈ C ∞ (R). After using integration by parts twice, we see that , for ξ = 0.

Expression (3.9) follows from the last equation and from the fact that ĥ
denotes the space of all complex-valued functions, which are absolutely continuous on R. Therefore, it follows from Lemma 3.1 above that if f ∈ W 2,1 (R), then f ∈ L 1 (R) ∩ C ∞ (R) and Remark 3.4. Suppose now that t ∈ (0, 1). Since The next three lemmas are elementary calculus results which will be used in the sequel.
Let x ∈ R. Since W 1,1 (R) ⊂ AC(R), we see that where the last expression is a consequence of the dominated convergence theorem.
The following five lemmas provide more explicit estimates than the corresponding results mentioned in [1]. The next lemma gives an upper bound, which goes to infinity as t tends to 1, for G(·, 1 − t 1/3 ) L 1 when t ∈ [0, 1).
Lemmas 3.6 and 3.9 below provide estimates for K(·, t) L 1 and ∂ x K(·, t) L 1 , for any t > 0. where C is a positive constant independent of t.
The following result gives an upper bound for ∂ x K(·, t) L 1 when t ∈ (0, 1). and

19)
where C is a positive constant independent of t.
Next lemma will be useful to study ∂ x K(·, t) L 1 for t ≥ t 0 , where t 0 > 0.

21)
where C is a positive constant independent of t.
The next result provides a unified upper bound for ∂ x K(·, t) L 1 for any t > 0, which takes the best of the corresponding bounds obtained in Lemmas 3.7 and 3.8. Lemma 3.9. Suppose that t > 0. Then the function ∂ x K(·, t) ∈ L 1 (R) ∩ C ∞ (R). Moreover,

22)
where C is a positive constant independent of t.
The following lemma will be used in the proof of Lemma 3.11 below. Proof. We recall that This concludes the proof.
The next lemma shows that (E(t)) t≥0 is a C 0 -semigroup on the Banach space Y and also on the Banach space X. Lemma 3.11. i.) If u 0 ∈ C b (R), then u(t) := E(t)u 0 ∈ C b (R) for every t ≥ 0. In addition, K(x − y, t)u 0 (y)dy, it follows that where the last inequality is a consequence of Lemma 3.6. Moreover, Thus, we have proved that if u 0 ∈ C b (R), then u(t) ∈ C b (R) for all t ≥ 0. In addition, one can see that E(t + s)φ = E(t)E(s)φ, for all t, s ≥ 0, and φ ∈ C b (R).
Suppose now that t = 0, u 0 ∈ Y \ {0}, and h ∈ (0, 1). SinceK(0, h) = 1 √ 2π K(z, h)dz = 1, and using (3.11) we have that Let ǫ > 0. By Lemma 3.10, there exists A > 0, such that for every h ∈ [0, 1), Since u 0 is uniformly continuous, there exists δ > 0 such that for all z, w ∈ R, for all x ∈ R, where the last inequality is a consequence of Young's inequality. Therefore, lim We notice that if u 0 ∈ Y , then u(t) = E(t)u 0 ∈ Y for all t ≥ 0. In fact, assume t > 0 and let ǫ > 0 be given. Since u 0 is uniformly continuous, there exists δ > 0 such that if |h| < δ, then |u 0 (x + h) − u 0 (x)| < ǫ √ 2π/ K(·, t) L 1 , for any x ∈ R. Suppose |h| < δ, then Hence, u(t) is uniformly continuous, for all t > 0. Assume now that t > 0, and u 0 ∈ Y . It follows from (3.28) and the semigroup property that lim On the other hand, for h > 0, one can see that where the last inequality is a consequence of Lemma 3.6. Equation (3.28) and the last inequality imply that ii.) Let u 0 ∈ X. By item i.) above, we already know that u ∈ C([0, +∞); Y ), where u(t) = E(t)u 0 for all t ≥ 0. We will now prove that ∂ The dominated convergence theorem and Lemma 3.6 imply that the last expression tends to zero as h goes to zero. Then there exists for all x ∈ R, and t > 0. (3.29) It is easy to see that the last expression is also valid if we only require that u 0 ∈ C 1 b (R). It follows from (3.29) and Lemma 3.6 that ∂ x u(·, t) L ∞ ≤ C · 1 + t 2 e 4 27 a 3 t u ′ 0 L ∞ , for all t > 0. Using the fact that u ′ 0 is uniformly continuous, (3.29), and Lemma 3.6, it follows that ∂ x u(·, t) is uniformly continuous, for all t > 0.
Finally, since (E(t)) t≥0 is a C 0 -semigroup on the space Y and using (3.29), we see that lim h→0 ∂ x u(t + h) − ∂ x u(t) L ∞ = 0, for all t ≥ 0.

3.2.
Local Theory in the Space X. In this Sub-section we will use the Banach fixed-point theorem on an appropriate complete metric space to find a local-in-time solution to the integral equation associated to the IVP (3.1). The following lemma will be helpful during the proof of Theorem 3.1 below.
Proof. i.) Let t ∈ (0, T ]. Now we first prove that D(t) ∈ X . In fact, Using the fact that ∂ x u(·, s)u(·, s) is uniformly continuous on R, for all s ∈ [0, T ], Lemma 3.6, and the dominated convergence theorem, it follows from the last inequality that D(t) is uniformly continuous on R. Now we claim that there exists ∂D ∂x (x, t) (in the classical sense), for all x ∈ R, and ∂D ∂x ( We now establish the last claim. It follows from Lemmas 3.6, 3.9, and 3.4 that K(·, t − s) * 1 2 ∂ x u 2 (·, s) ∈ C 1 (R) ∩ W 1,∞ (R), and for all x ∈ R and s ∈ [0, t). Moreover, In addition, where the last inequality is a consequence of Lemma 3.9. The claim now follows from (3.34)-(3.36) and the dominated convergence theorem. It follows directly from (3.33) and Lemma 3.9 that The fact that ∂ x D(·, t) is uniformly continuous on R can be shown similarly to the analogous result for D(·, t), using Lemma 3.9 instead of Lemma 3.6.
ii.) We will now prove that D ∈ C([0, T ]; X). Let t ∈ [0, T ). We first assume that h > 0. Then We see that where the last inequality follows from Lemma 3.6 and the fact that h ∈ (0, T ). Thus, using Lemma 3.11 and the dominated convergence theorem we have that Moreover, using Lemma 3.6 we get like in (3.31) that On the other hand, it follows from (3.33) that It follows directly from Lemma 3.9 that given by (3.38). To estimate J 1 (t, h) we first extend ∂ x K for all times in the following way: We note that H ∈ L 1 (R 2 ). In fact, by Lemma 3.9 we get |H(x, s)|dxds = T 0 ∂ x K(·, s) L 1 ds ≤ C µ(T ). Then where the last assertion follows from the continuity of translations in L 1 (R 2 ). Hence, lim where E(t) is defined by (3.6).